Theorem tsna1

Description: A Tseitin axiom for logical incompatibility, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018)

	Ref	Expression
Assertion	tsna1	$$⊢\theta \to \left(\left(\neg \phi \vee \neg \psi \right)\vee \neg \left(\phi ⊼\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		tsan1	$$⊢\theta \to \left(\left(\neg \phi \vee \neg \psi \right)\vee \left(\phi \wedge \psi \right)\right)$$
2		notnotb	$$⊢\left(\phi ⊼\psi \right)\leftrightarrow \neg \neg \left(\phi ⊼\psi \right)$$
3		df-nan	$$⊢\left(\phi ⊼\psi \right)\leftrightarrow \neg \left(\phi \wedge \psi \right)$$
4	2 3	bitr3i	$$⊢\neg \neg \left(\phi ⊼\psi \right)\leftrightarrow \neg \left(\phi \wedge \psi \right)$$
5	4	con4bii	$$⊢\neg \left(\phi ⊼\psi \right)\leftrightarrow \left(\phi \wedge \psi \right)$$
6	5	orbi2i	$$⊢\left(\left(\neg \phi \vee \neg \psi \right)\vee \neg \left(\phi ⊼\psi \right)\right)\leftrightarrow \left(\left(\neg \phi \vee \neg \psi \right)\vee \left(\phi \wedge \psi \right)\right)$$
7	1 6	sylibr	$$⊢\theta \to \left(\left(\neg \phi \vee \neg \psi \right)\vee \neg \left(\phi ⊼\psi \right)\right)$$